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Intrinsic diophantine approximation in Carnot groups and in the Siegel model of the Heisenberg group

10$
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Intrinsic diophantine approximation in Carnot groups and in the Siegel model of the Heisenberg group

10$

Anton Lukyanenko & Joseph Vandehey 

Abstract

We study an intrinsic notion of Diophantine approximation on a rational Carnot group G. If G has Hausdorff dimension Q, we show that its Diophantine exponent is equal to (Q+1)/Q, generalizing the case G=\mathbb {R}^n. We furthermore obtain a precise asymptotic on the count of rational approximations. We then focus on the case of the Heisenberg group \mathbf {H}^n, distinguishing between two notions of Diophantine approximation by rational points in \mathbf {H}^n: Carnot Diophantine approximation and Siegel Diophantine approximation. We provide a direct proof that the Siegel Diophantine exponent of \mathbf {H}^1 is equal to 1, confirming the general result of Hersonsky-Paulin, and then provide a link between Siegel Diophantine approximation, Heisenberg continued fractions, and geodesics in the Picard modular surface. We conclude by showing that Carnot and Siegel approximation are qualitatively different: Siegel-badly approximable points are Schmidt winning in any complete Ahlfors regular subset of \mathbf {H}^n, while the set of Carnot-badly approximable points does not have this property.

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Year 2020
Language English
Format PDF
DOI 10.1007/s00605-020-01406-7